Densities of heterogeneous random walks in ordered lattices. Random walk
Student Vasily marks the passed session. He walks through the city center, the diagram of which is shown in the figure. Being at a crossroads, Vasily randomly decides where to go next (he chooses each of the four possible directions with probability 1/4). If he finds himself next to a bar (green circles in the picture), he will certainly go into it, and if he finds himself next to a metro station (red circles in the picture), he will decide that this is a sign of fate and it’s time for him to stop having fun and need to go home. How find the probability that Vasily will get to the bar and continue his holiday if he is standing at an intersection marked with a blue circle?
Clue
It turns out that calculating the probability of getting to the bar from one specific intersection is inconvenient (if not impossible). It is better to calculate such probabilities for all intersections at once. For some intersections, the probability is known from the condition: if Vasily happens to be at a bar, then with probability 1 he will go into it, and if he happens to be at the metro, he will go home, and thus enter a bar with probability 0. It remains to understand how the desired probabilities for neighboring intersections are related to each other.
Solution
As stated in the hint, we will look for the probability of getting to the bar for all intersections at once. Let's denote these probabilities as shown in the figure on the right (and we'll call the intersections themselves the same for brevity). How are the probabilities of neighboring intersections related? The answer is given by the total probability formula: the probability of getting to the bar from a given intersection is equal to the arithmetic average of the probabilities of neighboring intersections (here we took into account that Vasily chooses equally likely). Let's say a = (b+ 1)/4, because this intersection has neighboring probabilities b, 1, 0 and 0, and d = (b + e + f)/4, because it has neighboring probabilities b, e, f and 0.
You can guess such a connection without knowing the formula for total probability, but by actually deriving it through the process of reasoning. Using the example of an intersection d it looks like this: with a probability of 1/4, Vasily will go to the intersection b, from which with probability b will end up in a bar, that is, this scenario will lead Vasily to the bar with a probability b·1/4. Similarly, we find the probabilities in the other three options, and their sum should be equal to d.
Thus, we obtain a system of linear equations
solving which, we get:
So, from his starting position, Vasily will hit the bar with a probability of 38/91. Good chances!
Afterword
This problem can also be solved empirically by writing a program that will imitate the behavior of our Vasily. By running this program many times, you can calculate in what proportion of experiments virtual Vasily gets to the virtual bar, and get an approximation to the correct answer. This approach is called the Monte Carlo method (there was already a problem on “Elements” illustrating this method).
You can use the “programmer” approach in another way: assign the required probabilities a, b, c, d, e, f some initial values, and then iteratively correct them using the equalities from the system that resulted in the solution. Straight ahead: adjust first a, Then - b, substituting the new value A, Then - With, ... After adjustment f, you need to repeat this cycle again, and so on, until the desired accuracy is achieved. This is a relaxation method.
Random walks on a plane and in space are convenient models for many physical, economic and biological processes. Here are just a few examples: Brownian motion and diffusion of substances, genetic drift, eye movements. More examples and properties of random walks can be found on Wikipedia.
If we calculate (in the same way) in our problem the probability of getting hit from an intersection d not in a bar, but in the subway, you get 53/91. Which in sum with 38/91 gives 1. That is, the probability that Vasily will wander around the city indefinitely is 0. This does not mean that he cannot just walk at all (for example, he can walk back and forth between intersections A And b), but such options very little compared to all possible routes of Vasily. This is true for any starting location and for any limited city with a similar “square” layout: making equally probable random decisions at each intersection, sooner or later Vasily will end up either in a bar or in the subway. This is a variation of the problem of ruining a player (see gambler's ruin) - just monitor the horizontal displacement.
It's interesting to see what happens with a random walk if there are no restrictions on the size of the region. For example, in the one-dimensional case, wandering occurs along a straight line on which points are marked at equal intervals, which are numbered in order with integers. The tramp starts at zero, and before each move he flips a coin to decide which way to go, after which he moves to an adjacent point in the chosen direction. It turns out that with probability 1 he will return to where he started. This is Polya's theorem for the one-dimensional case, it is not difficult to prove (the proof is given in small print).
First, let's consider the final version: wandering occurs on a segment with endpoints 0 and n. Let's denote by p(x) the probability that the walk will arrive at 0 earlier than at n. Then this function (defined on the set of integers from 0 to n) satisfies the following properties: (1) p(0) = 1, p(n) = 0; (2) p(x) = (p(x− 1) + p(x+ 1))/2. The second property is a sign of arithmetic progression, so the function values p(x) must form an arithmetic progression. From here and from condition (1) it follows that p(x) = 1 − x/n.
Let it now P(n) - the probability that a tramp, wandering along an already infinite straight line, will return to the beginning before he moves away from him n. For example, P(1) = 0 (because on his first move he will move away from the beginning by 1), and P(2) = 1/2, because from point 1 s equal probability you can go back to the beginning and step into 2, and the same is true for point −1. Let's show that P(n) = 1 − 1/n. Let the tramp move to 1 on his first move. Then the case discussed above gives us that with probability 1 − 1/ n he will arrive at 0 before he reaches the point n. Likewise, if he moved to point −1 on his first move. That's why P(n) = (1 − 1/n)/2 + (1 − 1/n)/2 = 1 − 1/n. That's what was required.
In the two-dimensional case, Polya’s theorem is also true: a random walk along the lines of an infinite checkered grid with probability 1 will return to starting point. This is, however, more difficult to prove than the one-dimensional case. Surprisingly, in three-dimensional space (and in higher dimensions) this theorem is no longer true. Beautiful proof of these facts is based on an unexpected analogy between random walks and electrical circuits. You can read about this in detail in the article by M. Skopenkov, V. Smykalov and A. Ustinov “Random walks and electrical circuits” (published in the collection “Mathematical Enlightenment”, issue 16), on the basis of which this problem was prepared.
There is another interesting problem, the solution of which cannot be done without the concept of probability. This is the "random walk" problem. In its simplest form, this task looks like this: Imagine a game in which a player, starting from point x = 0, can move either forward (to point x) or backward (to point -x) in each move, and the decision about where to go is made completely randomly, well, for example, by tossing a coin. How to describe the result of such a movement? In more general form this problem describes the movement of atoms (or other particles) in a gas - so-called Brownian motion - or the generation of errors in measurements. You will see how the random walk problem is closely related to the coin toss experiment described above.
First, let's look at some examples of random walks. They can be described as “pure” promotion D N, in N steps. On fig. 6.5Three examples of random walk paths are shown.
What can be said about such a movement? Well, first of all, one might ask: How far will we get, on average? We must expect that there will be no average progress at all, since we are equally likely to go forward as well as backward. However, it feels like as N increases we are increasingly likely to wander somewhere further and further from the starting point. The question therefore arises: what is the average absolute distance, i.e., what is the average value of |D|? However, it is more convenient to deal not with |D|, but with D 2 ; this quantity is positive for both positive and negative motion and therefore can also serve as a reasonable measure of such random walks.
It can be shown that the expected value of D 2 N is simply equal to N-the number of steps taken. By the way, by “expected value” we mean the most probable value (guessed the best way), which can be thought of as the expected mean large number repeating processes of wandering. This value is denoted as
Expected value D 2 N for N > 1 can be obtained from D N-1 . If after (N - 1) steps we are at a distance D N-1 then one more step will give either D N = D N -1 + 1, or D N =D N -1 - 1. Or for squares
If the process is repeated a large number of times, then we expect each of these possibilities to occur with probability 1/2, so that the average expected value will simply be the arithmetic mean of these values, i.e., the expected value D 2 N will be just D 2 n-1 + 1. But what is the value of D 2n-1 , or rather, what value do we expect? Simply, by definition, it is clear that this should be the “average expected value”
If we now remember that
Deviation from initial position can be characterized by a quantity of the type of distance (rather than the square of the distance); to do this you just need to extract Square root from D< 2 N > and get the so-called “mean square distance” Dсk:
We have already said that random walks are very similar to the coin toss experiment with which we began this chapter. If we imagine that every move forward or backward is determined by the appearance of “heads” or “tails”, then D N . will simply be equal to N 0 - N P , i.e. the difference in the number of “heads” and “tails” occurrences. Or since N 0 + N P = N (where N is the total number of tosses), then D N = 2N 0 - N. Remember that earlier we already received an expression for the expected distribution of the value No [it was then denoted by k; see equation (6.5)]. Well, since N is simply a constant, now the same distribution has turned out for D. (The loss of each “heads” means the non-falling of “tails”, therefore, in the connection between N 0 and D a factor of 2 appears.) Thus, in FIG. 6.2, the graph simultaneously represents the distribution of distances that we can go in 30 random steps (k = 15 corresponds to D = 0, and k = 16 corresponds to D = 2, etc.).
Deviation N 0 from the expected value N/2 will be equal
whence for the standard deviation we obtain
Let us now recall our result for D sk . We expect that the average distance covered in 30 steps should be √30 = 5.5, whence the average deviation of k from 15 should be 5.5: 2 ≈ 2.8. Note that the average half-width of our curve in Fig. 6.2 (i.e., the half-width of the “bell” is somewhere in the middle) is approximately equal to 3, which is consistent with this result.
We are now able to consider a question that we have avoided until now. How do we know if our coin is “fair”? Now we can, at least partially, answer it. If the coin is “fair”, then we expect that half the time it will come up “heads”, i.e.
At the same time, it is expected that the actual number of heads should differ from N/2 by an amount of the order of √N/2, or, if we talk about the fraction of deviation, it is equal to
i.e., the larger N, the closer to half the ratio N0/N.
On fig. 6.6numbers N are set aside 0
/N is for those coin tosses we talked about earlier. As you can see, as N increases, the curve gets closer and closer to 0.5. But, unfortunately, there is no guarantee that for any given series or combination of series the observed deviation will be close to the expected deviation. There is always a finite probability that there will be a large fluctuation—a large number of heads or tails—that will produce an arbitrarily large deviation. The only thing that can be said is that if the deviations are close to the expected 1/2√N (say, by a factor of 2 or 3), then there is no reason to consider the coin to be “counterfeit” (or that the partner is cheating).
We have not yet considered cases when for a coin or some other test object similar to a coin (in the sense that two or more reliably unpredictable outcomes of an observation are possible, for example a stone that can only fall on one of two sides) , there is enough reason to believe that the probabilities of different outcomes are not equal. We defined the probability P(O) as the ratio
This notation implies that there is a certain “true” probability, which in principle can be calculated, but that various fluctuations lead to an error in its experimental determination. However, there is no way to make these arguments logically consistent. It is better, after all, for you to understand that probability in some sense is a subjective thing, that it is always based on some uncertainty of our knowledge and its value fluctuates as it changes.
WHAT THE HELL IS THIS “RANDOM WALKING”?
Chartism is as old as Egyptian papyri. The random walk method also has ancient roots, but in its finished form is as young as computers. Chartism tries to find some order in what is happening - the “random walk” method claims that there is no order. And if the proponents of the random walk theory are right, then the chartists are about to be out of work, and menacing clouds have gathered over all securities analysts.
Proponents of the “random walk” are mostly university professors working in the faculties of business and economics. They have a good command of complex mathematical language and enjoy using it. Moreover, articles on “random walk” written by these scientists simply have to be completely incomprehensible to the uninitiated and oversaturated with mathematical symbols in order to make the proper impression on their colleagues. If you want to see what it looks like, try reading the magazine “Kyklos” - there are more than one or two such articles in it. Extensive material related to the topic of interest to us can be found there. But we will find it in the collection “The Random Nature of Prices in the Stock Exchange” (published by the Massachusetts Institute of Technology, edited by Professor Paul Kutner), and in the 16th issue of “Selected Papers of the Faculty of Business of the University of Chicago,” in the work of Professor Eugene Feima “ Random walk applied to stock market prices.”
What is a "random walk"? I am not able to understand half the articles on this subject, since my knowledge of Boolean algebra is limited and my knowledge of stochastic series is zero. But after a series of conversations with random walk guys, it dawned on me that the whole trick can be summed up in one single sentence. Later, Professor Kutner, through one of my friends, conveyed that my department was quite suitable, and therefore, without any equations, S and D, I present it here.
Prices have no memory, and yesterday has nothing to do with tomorrow. Each new day begins with a 50/50 probability. Yesterday's prices already included all the details of yesterday. Or, as Professor Feima said, " past history series (of changes in stock price) cannot be used to predict the future in any rational way. The future movement of the price level as a whole or the price of an individual asset is no more predictable than the movement of a series of random numbers.”
Of course, it is not only university professors who are engaged in disorder as a way to beat the market. Senator Thomas J. McIntyre, a New Hampshire Democrat and member of the powerful Senate Banking Committee, one day brought with him an ordinary wall-mounted dartboard. He attached to it a list of companies from the stock exchange and began throwing darts. The stock dart selection outperformed the vast majority of mutual fund portfolios. (Thus, Senator McIntyre's darts confirmed the testimony of the random walk theorists, Professors Paul Samuelson of MIT and Henry Wallich of Yale, given at the Senate hearings discussing mutual fund legislation.) If such big guns as Professors Samuelson and Wallich, plus banking Senate committee takes the “random walk” so seriously, then everyone else should think hard: after all, if the “random walk” really is the Truth, then the value of all charts and all investment advice is zero - and this can very seriously affect the rules of the Game.
Do you know that: broker binary options Binomo regularly conducts for its clients tournaments With free registration and real cash prizes.
The first assumption of the "random walk" is that a market - such as the New York Stock Exchange - is an "efficient" market, that is, one where the numbers are rational and profit-seeking investors compete with each other with roughly equal access to information and trying to determine future price behavior.
The second basic thesis is that stocks have intrinsic value - an "equilibrium price" in economist's parlance - and that at any given moment the price of a stock can be a good indicator of its intrinsic value, which generally depends on the profitability of that stock. But since no one can say with certainty what the actual value is, then, as Professor Feima says, “the actions of many competing participants should cause the current price of a stock to randomly wander around its actual value.”
Proponents of the "random walk" have tested their theory against "empirical evidence." The purpose of the study was to demonstrate mathematically that successive price changes occur independently of each other. Here's a fragment of one of the texts - just to really scare you. Its author was MIT professor William Steiger, and the work itself was published in the collection “The Random Nature of Prices on the Stock Exchange.”
“The test is based on the sampling distribution of statistics related to purely random walks, the nature of which I formulated earlier. Assuming that t is the ratio (random variable) of the range of deviation from the straight line connecting the first and last values segment of a continuum random walk to the sample standard deviation of the increment, this distribution determines the probability P„where t is less than or equal to any t.
Let's consider the following stochastic process. Let's assume that
In case you didn't know this before, we're talking about about serial correlation coefficients - and looking at them, I experience the same feeling as you. Another approach to the problem is to test mechanical trading rules to see if they produce better results than simply buying and holding stocks. Professor Sidney Alexander of MIT, for example, tried all sorts of filters, using the test results to conclude what would happen if different mechanical bidding rules were followed.
(The 5 percent filter works like this: If a stock goes up 5 percent on any given day, buy it and hold it until the price changes from last highest point will not move down 5 percent. Then you should sell them and then go short selling. Continue short selling until the closing price is at least 5 percent higher than the last low. In this case, cover what you sold and start buying.)
As you can see, the filter is really about trend analysis and measuring price movements. Professor Alexander reports testing of filters ranging from 1 to 50 percent (see Price Movements in Speculative Markets: Trends and Random Walks). It turned out that simply buying and holding stocks consistently gave better results than using any of the filters.
Therefore, proponents of the random walk argue that a statement such as “a stock with an established trend is more likely to continue moving with that trend” is absolute nonsense. The chances of whether the stock's trend will continue or not are fifty-fifty.
The same can be said about tossing a coin. If you flip a coin five times and it comes up heads five times in a row, what are the chances that it will come up open the sixth time? If you toss a coin a hundred times and it comes up heads a hundred times in a row, what are the chances that it will come up heads the hundredth time? The same fifty-fifty.
“If the random walk model adequately describes reality,” says Professor Feima, “then the work of a technical analyst, like the work of an astrologer, has no real value.”
Proponents of random walk are especially aggressive towards chartists. As I've said before, one random walk professor literally choked on his dessert in my house when someone dared to say that maybe diagrams should be taken seriously. (It is now a rule in our family that all random walk proponents must finish their dessert before the topic of graphs and diagrams can be broached.) Another professor I know, a random walk apologist, began tossing a coin with his students, mistaking heads for pluses and tails for minus. Then they made a chart, putting a cross when it came up on heads and a zero when it came up on tails. And what do you think? The result is a classic tic-tac-toe chart, with all the essential elements: “head and shoulders”, “ reverse movements", "double tops" and all the rest.
But proponents of the random walk don't stop at attacking the Chartists. They intend to seriously disturb fundamentalist analysts as well. This is how they reason in this case.
There are discrepancies between the actual price and the actual intrinsic value of a stock. The analyst collects all the information available to him and, using all his knowledge and talents, speaks out for a purchase or, accordingly, a sale. His actions help narrow the existing gap between price and intrinsic value. And the better and more sophisticated the analysts, the more to a greater extent they neutralize themselves because the market becomes more and more “efficient.” And an “efficient” market is clearly consistent with a random walk model, where intrinsic value is already taken into account and reflected in the price.
It is clear that an analyst who is one step ahead of the rest, in an efficient market, will exceed the total average result of his colleagues, but the thing is that all analysts are convinced that their abilities and professionalism are above average. The analyst's performance should consistently be higher than the performance of a randomly selected portfolio of assets of the same nature, if only because each analyst has a 50 percent chance of outperforming the random sample, even if he is a complete idiot or uses a dartboard instead of a slide rule.
The world of random walk is a cold, harsh and very negative world. Adherents of this theory believe in the existence of the intrinsic value of a stock, but this does not make it any easier for us, because stocks are sold at their intrinsic value - whatever we understand by this term - only at those moments when the market crosses this mark, moving up or down. In other words, intrinsic value turns out to be the correct reference point in the same sense in which a stopped clock shows right time twice a day.
As we already know, there are eleven thousand securities analysts - and, of course, many thousands of chartists. Chartists don't believe in random walk because such a belief would make their work meaningless - what kind of professional enjoys knowing that the dartboard is working as efficiently as he is? As for analysts, they believe that random walk plays no role because their awareness and intuition allow them to be ahead. Neither of them seriously dives into the mathematical proof of random walk theory. If they did this and accepted the arguments presented, they might accept some loss of salary and switch to teaching in business schools, but so far there has been no significant outcome in this direction.
In support of the skeptics, we can only once again appeal to the premise that the exchange is reasonably “efficient”, that is, that it is a market where the numbers are rational and profit-oriented investors compete with each other. It is quite likely, however, that investors - and even cold, stern, professional investment managers - are not rational, or not 100 percent rational. Perhaps they prefer to have some profit and feel; that they are not alone in their decisions than to have maximum profit and experience constant anxiety. An investor in a random walk model behaves with suspicious consistency as a “homo economicus”, and we have already argued more than once that “homo” is not quite an “economicus” after all. As Lord Keynes said, “There is nothing more disastrous than a rational investment strategy in an irrational world.”
So far, no one has been able to squeeze emotions into serial correlation coefficients and serial test run analysis. It is absolutely true that, statistically speaking, tomorrow's stock price has no relation to its yesterday's price. But people, the Crowd, are endowed with a memory that covers both that day and this one. You've probably noticed something that's as common to the world of random walk as it is to the world of graphs and charts: neither world has room for people. There are prices, there are odds, there is a past (or there is none - depending on which of the two theories you adhere to). Bishop Berkeley's tree falls in the forest and makes a terrible noise, although there is no one to hear the noise.
If the stock exchange is really a Game, then the Game can be played without any internal values. And if one of the rules of the Game says that Bishop Berkeley's tree falls when everyone decides that it has fallen, then there is not even a need for the tree itself. If printers continue to print stock certificates, the New York Stock Exchange continues to open, and banks continue to print dividend figures from time to time, then the whole Game remains in place, even if all the steel mills, warehouses and railways mysteriously disappeared - provided that none of the participants in the Game knows about it.
Proponents of random walk turn to computers for more complex proofs of the correctness of their theory, hoping to gain additional power and authority. Technical analysts also turn to computers, running selections and filters set not only on closing prices, but also on highs and lows, moving averages, etc. - in general, according to any conceivable serial ratio of quantities. But computers are programmed by people; machines cannot think for themselves. Therefore, the same computers do not produce the same evidence. The first challenge to the mathematical language of random walk theory was posed in Roberge Levy's The Concept of Relative Strength - and an answer to it in the same language is probably brewing somewhere.
The influence of random walk theory should be beneficial by definition simply because it forces everyone to check and double-check their results instead of accepting myths and generalizations on faith. But at the same time - I am not hinting at anything here - there are very few rich people among the adherents of the random walk, just as there are few of them among the chartists. On the other hand, there are very successful investors who do not have any formulated systems. Maybe they just got into a lucky streak of trades, maybe they're more rational or have better access to information. Or maybe they don’t want to take this into account either. harsh world statisticians are simply better experts in human psychology.
Proponents of the random walk do not unanimously claim that the stock exchange is a random walk. Some admit: no, this is not entirely true - if only because the market is far from perfect, from complete “efficiency”. In other words, because there are people on it. “My model,” writes Professor Kutner, “is entirely consistent with how I see chart reading on Wall Street. Like the Indian healers who discovered tranquilizers, Wall Street shamans, without any scientific methods, with the help of their magic they still produce something, having no idea what they produced and how it works.” And Professor Alexander concludes one of his papers this way: “In a speculative market, price appears to follow the principle of a random walk over time, but its movement, once begun, tends to continue.”
But based on the movement, which tends to continue, it is already possible to construct a diagram. (“The results of statisticians in the study of random walk over a long time interval do not contradict non-random trends in the interval of movement that occurs,” writes Professor Alexander.)
To be fair, you should apply the bias already mentioned in this book to both the charts and the random walk. We briefly touched on the diagrams, but the technical work covers, in addition to price movements, other factors (sales volume, its growth, decline, etc.), which the diagrams readily demonstrate to us. My bias, which I have already admitted, is a love of “cumulative earnings,” which fits neatly into the old fundamentalist concept called “Present Value of Future Earnings.” And it’s just a stone’s throw away from the classic fundamentalist theory of “The Present Value of Future Dividends.” There is no doubt that the idea of Intrinsic Value is woven into growing earnings, but the Game can be played with Intrinsic Value. And if the stock exchange is a Game, then the attempts of statisticians to destroy charts and graphs are not at all as terrible as they seem. The Chartists, taken together, are becoming a serious market force in their own right. Maybe they just belong to the irrational and as yet unmeasured australopithecus side of the market.
There is another complaint that should be made against academic researchers: they tend to give lectures in a language that the listener does not speak, such as the language of quadratic equations. “There is a peculiar paradox in the relationship between mathematics and the investor's attitude toward stocks,” writes Benjamin Graham, doyen financial analysis, in his book The Intelligent Investor. Graham continues:
“Mathematics is believed to produce accurate and reliable results. But in the stock market, the more sophisticated and complex the mathematics, the more unreliable and speculative the conclusions we draw from them. In all my forty-four years of experience on Wall Street, I have never seen a reliable calculation of stock value or related investment strategy that went beyond simple arithmetic or the most basic algebra. If mathematical analysis or higher algebra comes into play, this is always a sign that the author is trying to replace experience with theory.”
As you might expect, given my own bias, I readily agree with the doyen of financial analysis here. Moreover. It seems to me that even if the adherents of the random walk were to announce that they had found an impeccable mathematical proof of the random nature of stock exchange processes, I would still continue to believe that in the long term future profits influence the current price, and in the short term they are the dominant factor What will remain is the elusive Australopithecus - the character and mood of the crowd.
There is another interesting problem, the solution of which cannot be done without the concept of probability. This is the "random walk" problem. In its simplest form, this task looks like this: Imagine a game in which a player, starting from point , for each move can move either forward (to point ) or backward (to point ), and the decision about where to go is made completely randomly, well, for example, using coin toss. How to describe the result of such a movement? In a more general form, this problem describes the movement of atoms (or other particles) in a gas - so-called Brownian motion - or the generation of errors in measurements. You will see how the random walk problem is closely related to the coin toss experiment described above.
First, let's look at some examples of random walks. They can be described as “pure” progress in N steps. In fig. Figure 6.5 shows three examples of random walk paths. (When constructing them as a random sequence of decisions about where to take the next step, the results of the coin toss shown in Fig. 6.1 were used.)
What can be said about such a movement? Well, first of all, one might ask: How far will we get, on average? We must expect that there will be no average progress at all, since we can go both forward and backward with different probabilities. However, it feels like as we increase we are increasingly likely to wander somewhere further and further from our starting point. Therefore, the question arises: what is the average absolute distance, i.e., what is the average value of ? However, it is more convenient to deal not with, but with; this quantity is positive for both positive and negative motion and therefore can also serve as a reasonable measure of such random walks.
Figure 6.5. Three examples of random walk.
The number of steps is plotted along the horizon, and the coordinate is plotted vertically, i.e., the net distance from the starting point.
It can be shown that the expected value is simply the number of steps taken. By the way, by “expected value” we mean the most probable value (best guessed), which can be thought of as the expected average value of a large number of repeated wandering processes. This value is designated as and is also called the “mean square of the distance.” After one step it is always equal to , therefore, undoubtedly, . (One step will be chosen throughout for a unit of distance, and therefore I will not write units of length in what follows.)
The expected value for can be obtained from . If after steps we are at a distance, then one more step will give either or. Or for squares
(6.7)
If the process is repeated a large number of times, then we expect that each of these possibilities occurs with probability , so that the average expected value will be simply the arithmetic mean of these values, i.e., the expected value will be simply . But what is the value, or rather, what value do we expect? Simply, by definition, it is clear that this should be the “average expected value”
(6.8)
If we now remember that , we get a very simple result:
The deviation from the initial position can be characterized by a quantity of the type of distance (rather than the square of the distance); to do this you just need to take the square root of it to get the so-called “mean square distance”
(6.10)
We have already said that random walks are very similar to the coin toss experiment with which we began this chapter. If we imagine that each advancement forward or backward is determined by the appearance of “heads” or “tails,” then it will simply be equal to, i.e., the difference in the number of occurrences of “heads” and “tails.” Or since N (where is the total number of tosses), then 
. Remember that earlier we already obtained an expression for the expected distribution of the quantity [it was then denoted by ; see equation (6.5)]. Well, since it is simply a constant, now the same distribution has turned out for . (The appearance of each “heads” means the failure of “tails”, so a factor of 2 appears in the connection between and.) Thus, in FIG. 6.2, the graph simultaneously represents the distribution of distances that we can go in 30 random steps (matches, matches, etc.).
The deviation from the expected value will be equal to
whence for the standard deviation we obtain
.
Recall now our result for We expect that the average distance covered in 30 steps should be equal to , whence the average deviation from 15 should be . Note that the average half-width of our curve in Fig. 6.2 (i.e., the half-width of the “bell” is somewhere in the middle) is approximately equal to 3, which is consistent with this result.
We are now able to consider a question that we have avoided until now. How do we know if our coin is “fair”? Now we can, at least partially, answer it. If the coin is “fair”, then we expect that half the time it will come up “heads”, i.e.
At the same time, it is expected that the actual number of heads should differ from by an amount of the order of , or, if we talk about the fraction of deviation, it is equal to
,
i.e. the more, the closer to half the ratio.
Figure 6.6. The percentage of heads in a particular sequence of coin tosses.
In fig. 6.6 sets aside the numbers for those coin tosses that we talked about earlier. As you can see, as the number increases, the curve gets closer and closer to 0.5. But, unfortunately, there is no guarantee that for any given series or combination of series the observed deviation will be close to the expected deviation. There is always a finite probability that there will be a large fluctuation—a large number of heads or tails—that will produce an arbitrarily large deviation. The only thing that can be said is that if the deviations are close to expected (say, by a factor of 2 or 3), then there is no reason to consider the coin to be “counterfeit” (or that the partner is cheating).
We have not yet considered cases when for a coin or some other test object similar to a coin (in the sense that two or more reliably unpredictable outcomes of an observation are possible, for example a stone that can only fall on one of two sides) , there is enough reason to believe that the probabilities of different outcomes are not equal. We defined probability as a ratio. But what should we take as the value? How can you know what is expected? In many cases, the best that can be done is to count the number of heads in a large series of trials and take (the observed). (How can you expect anything else?) It must be understood, however, that different observers and different series of tests may give a different value from ours. It should be expected, however, that all these different answers will not differ by more than [if close to half]. Experimental physicists usually say that the "experimentally found" probability has an "error" and write it as
. (6.14)
This notation implies that there is a certain “true” probability, which in principle can be calculated, but that various fluctuations lead to an error in its experimental determination. However, there is no way to make these arguments logically consistent. It is better, after all, for you to understand that probability in some sense is a subjective thing, that it is always based on some uncertainty of our knowledge and its value fluctuates as it changes.
